Distributions and Fourier transform, examples of Banach spaces: Lebesgue spaces
and their duals, spaces of continuous functions and measures, Sobolev spaces. The
structure theorems of functional analysis and applications: The theorem of Hahn-
Banach, Banach-Steinhaus and the uniform boundedness principle, closed graph
and open mapping theorem. Spectral theory of compact operators. The Lemma of
Lax-Milgram and elliptic PDEs. Examples of bounded operators and applications
to PDE.